Quasi-Three-Dimensional Simulation of Viscoelastic Flow through a Straight Channel with a Square Cross Section

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1 Article Nihon Reoroji Gakkaishi Vol.34, No.2, 105~113 (Journal of the Society of Rheology, Jaan) 2006 The Society of Rheology, Jaan Quasi-Three-Dimensional Simulation of Viscoelastic Flow through a Straight Channel with a Square Cross Section Shuichi TANOUE, Tomohiro NAGANAWA, and Yoshiyuki IEMOTO Deartment of Materials Science and Engineering, University of Fukui Bunkyo, Fukui-shi, Fukui , Jaan (Received : January 5, 2006) When a viscoelastic fluid flows in a straight channel with a noncircular cross section, a secondary flow is develoed. We discussed the effect of flow characteristics on the secondary flow of the viscoelastic fluid through a straight channel with a square cross section by the viscoelastic flow simulation using the finite element method. The Phan-Thien Tanner (PTT) model was emloyed as the constitutive equation. The second normal stress difference N 2 contributes to the secondary flow motion of a viscoelastic fluid through a straight channel with a square cross section. The degree of the secondary flow exressed by the maximum streamline function ψ max deends on the recoverable shear for the second normal stress difference N 2 /2σ, where σ is shear stress. According to the distribution of N 2 on the cross section, the flow direction of the secondary flow deends on the ressure dro generated by N 2. The elongational flow characteristics also contribute to the secondary flow state. On the whole, the degree of the secondary flow increases with the strain-hardening. However, the contribution of elongational flow characteristics to the secondary flow is smaller than that of N 2 /2σ. Key Words: Square channel flow / Viscoelastic flow simulation / Finite element method / Secondary flow 1. INTRODUCTION The secondary flow is the flow in the direction erendicular to the main flow. When a Newtonian fluid or a urely viscous fluid flows through a straight channel, no secondary flow is develoed. However, when a viscoelastic fluid flows in the straight channel with a noncircular cross section, a secondary flow is develoed. It is well known that the second normal stress difference N 2 of a viscoelastic fluid contributes to the develoment of secondary flow of the viscoelastic fluid through the straight channel with a noncircular cross section. Townsend et al. 1) discussed exerimentally the secondary flow in ie with non-circular cross section. They concluded that the major factor for the develoment of secondary flow is N 2. After that, theoretical studies using the numerical simulation were carried out by several researchers. Gervang and Larsen 2) discussed the secondary flow in straight ducts with rectangular cross section by numerical simulation using the Criminale-Ericksen-Filbey (CEF) model. 3) Xue et al. 4) investigated numerically the attern and strength of secondary flow in rectangular ies by the finite volume To whom corresondence should be addressed. tanoue@matse.fukui-u.ac.j, Tel: , Fax: method using the Phan-Thien Tanner (PTT) model. 5) Debbaut et al. 6, 7) studied the secondary flow exerimentally by observing the two layer flow of the same viscoelastic fluids and numerically by the finite element method using the Giesekus model. 8) They obtained the good agreement of their numerical results with exerimental observations. On the other hand, the secondary flow was focused by the study of co-extrusion rocess because the secondary flow contributes to the encasulation that the less viscous layer tends to encasulate the more viscous layer. Even if two layers are non-elastic fluids, the encasulation will occur. 9, 10) When two layers are viscoelastic fluids, this henomenon will occur more owerfully because a viscoelastic fluid is easy to develo the secondary flow even if mono-layer flows. Dooley et al. 11) studied the effect of the fluid viscoelasticity on the layer thickness of multilayer coextruded structure. Takase et al. 12) investigated numerically the effect of N 2 of test fluid on the encasulation in coextrusion rocess by viscoelastic flow simulation using the finite element method. They indicated that the degree of encasulation increases with N 2 of test fluid. Sunwoo et al. 13) discussed the viscoelastic coextrusion rocess by three-dimensional numerical simulation. They showed that the test fluid with large magnitude of N 2 rotruded into the test 105

2 Nihon Reoroji Gakkaishi Vol fluid with small magnitude of N 2 around the symmetric lane. By the develoment of comuter technology and calculation technique, we can do more detail investigations for the secondary flow on the viscoelastic flow through a straight channel by the viscoelastic flow simulation. In this study, we tried to discuss the effect of shear and elongational flow characteristics on the secondary flow state of the viscoelastic fluid through a straight channel with a square cross section by the viscoelastic flow simulation. The aims of this study are 1) the investigations of the effect of shear and elongational flow characteristics on the secondary flow state, and 2) the consideration of the mechanism of a secondary flow motion of the viscoelastic fluid through a straight channel with a square cross section. 2. CALCULATION THEORY 2.1 Governing Equations In this study, an isothermal, steady creeing viscoelastic flow of incomressible fluids with high viscosity is assumed. It is also assumed that the inertia is negligible because a viscoelastic fluid having high viscosity such as a olymer melt are assumed as a focused fluid and we want to consider urely the effect of shear and elongational flow characteristics on the secondary flow. The equation of continuity and the momentum equation are described as follows; (1) (2) where v is a velocity vector, a ressure, an extra stress tensor. The Phan-Thien Tanner (PTT) model 5) was used as a constitutive equation of the viscoelastic fluid. The basic form of the extra stress is shown as the following equations. (6) The notations and are resectively the uer and the lower convected time derivative defined as the following equations. (7) (8) In this study, the 1-mode tye model having one relaxation time was used because it is easy to understand the effect of flow characteristics on the secondary flow. We selected the PTT model as the constitutive equation in this study. One of the reasons is that we can consider simly the effect of N 2 and elongational flow characteristics on the secondary flow by changing one material constant (ξ or ε) of the PTT model. The detail story is described later. 2.2 Flow Modeling and Calculation Method In this study, we consider the viscoelastic flow including the secondary flow through a straight channel with a square cross section. Figure 1 shows the schematic diagram of a flow model. We used the rectangular coordinate system (x, y, z axes) for the calculations. The main flow direction was defined by the x axis, and then the secondary flow occurs on the y-z lane. In this study, we assumed that the flow geometry of the cross section, i.e., the shae of cross section on the y-z lane is the same along the main flow direction. Therefore we can assume that the ressure dro ( / x) and other hysical values such as velocity and stress excet for ressure are constant in the main flow direction. In short, we solved the fully develoed (3) (4) where η 0 is a zero shear viscosity, s is a material constant, D is a rate of deformation tensor and v is a velocity gradient tensor. The suerscrit t denotes the transose of a tensor. For the PTT model, E is given by the following equation. (5) where λ is a relaxation time, ε and ξ are material constants. G is a relaxation modulus defined as the following equation. Fig. 1. Schematic diagram of the flow model. 106

3 TANOUE NAGANAWA IEMOTO : Quasi-Three-Dimensional Simulation of Viscoelastic Flow through a Straight Channel with a Square Cross Section flow to the main flow direction (to x axis) in this study. The calculation region was the one-quarter region of the whole cross section of the straight channel shown in Fig. 1. The equation of continuity, the momentum equation and the constitutive equation were simultaneously solved for velocity comonents v, ressure and stress comonents E, using the Galerkin finite element method. In order to calculate the flow fields at the high shear rate, we used the under-relaxation method for the rate of deformation tensor roosed by Tanoue et al. 14) In addition, we need to calculate the ressure dro ( / x) as one of the variables. In order to calculate the ressure dro, we added the following equation of the volume flow rate Q to the governing equations: (9) where A is the cross-sectional area of a channel. Figure 2 shows the boundary conditions. We imosed the non-sli conditions on the wall, the symmetric conditions on the symmetric lanes, the x-z lane at y = 0 and the x-y lane at z = 0, and the ressure at the corner of the channel is equal to zero. The nine-node Lagrangian quadrilateral element was used in this study. The velocity and stress comonents were interolated in terms of bi-quadratic Lagrangian olynomials, while the ressure was interolated in terms of bi-linear olynomials. A set of nonlinear algebraic equations was solved by the Newton-Rahson method. the material constants, ξ = 0.2 and ε = 0.08 was defined as the base fluid. This base fluid is assumed to be the LDPE melt. The material constants for test fluids were set out as follows. 1) In Case 1, we set the material constants of the PTT model on ξ = 0.05, 0.2 and 0.5 at ε = 0.08 and s = 1/9, for consideration of the effect of the second normal stress difference on a secondary flow state. The relationshi between the normal stress differences and ξ for the PTT model is exressed as the following equation; (10) where N 1 is the first normal stress difference. Therefore the contribution of N 2 increases with ξ. Figure 3 shows the steadystate shear and lanar elongational flow characteristics of the PTT model in Case 1. Other arameters are s = 1/9 and ε = Test Fluids In this study, we focused the two flow characteristics affecting the secondary flow state of viscoelastic fluid through a straight channel: 1) the second normal stress difference, and 2) the elongational flow characteristics. The fluid having Fig. 2. Boundary conditions. Fig. 3. Steady-state flow characteristics curves of the PTT model in Case 1 (ε = 0.08 and s = 1/9). (a) Dimensionless shear viscosity η/η 0 and dimensionless rimary normal stress difference N 1 /G versus dimensionless shear rate λγ, (b) Dimensionless secondary normal stress difference N 2 /G versus dimensionless shear rate λγ, (c) Dimensionless lanar elongational viscosity η EP /η 0 versus dimensionless lanar elongation rate λε. 107

4 Nihon Reoroji Gakkaishi Vol The absolute values of the second normal stress difference N 2 increase with ξ. We can simly discuss the effect of N 2 on the flow state by changing ξ at constant ε. The first normal stress difference N 1 increases with a decrease of ξ. 2) In Case 2, we set the material constants of the PTT model on ε = 0.013, 0.08 and 0.4 at ξ = 0.2 and s = 1/9, for the consideration of the effect of the elongational flow characteristics on the secondary flow state. Figure 4 shows the steady-state shear and lanar elongational flow characteristics of the PTT model in Case 2. The strain-hardening increases with a decrease of ε. However, the shear flow characteristics are almost indeendent of ε. Therefore, we can simly discuss the effect of elongational flow characteristics on the flow state by changing ε at constant ξ. In this study, we assume the viscoelastic flow through a die on extrusion molding. Therefore, the focused region of dimensionless shear rate λγ will be less than at most The ψ R is the streamline function with dimensional unit. We used the half length of a channel width as the reference length H. Figure 6 shows ψ max as a function of the Weissenberg number We in the case using three kinds of finite focused region of dimensionless lanar elongation rate λε may be less than 1 because the velocity of secondary flow would be much smaller than that of main flow. The actual region of λε that we should focus in this study will be shown later. 3. RESULTS AND DISCUSSION 3.1 Mesh Refinement Before the detail discussion of the secondary flow, we should determine the mesh density for the calculation. In this consideration, we used the test fluid having the material constants of ξ = 0.2, ε = 0.08 and s = 1/9. Figure 5 shows the finite element meshes used in this consideration. The 3 x 3, 5 x 5 and 10 x 10 meshes were used. We estimated the maximum absolute value of dimensionless streamline function for the secondary flow ψ max. This dimensionless streamline function ψ was defined as the equation, ψ = ψ R /(<u>h), where <u> is the mean velocity of a main flow, H is the reference length and Fig. 4. Steady-state flow characteristics curve of the PTT model in Case 2 (ξ = 0.2 and s = 1/9). (a) Dimensionless shear viscosity η/η 0 and dimensionless rimary normal stress difference N 1 /G versus dimensionless shear rate λγ, (b) Dimensionless secondary normal stress difference N 2 /G versus dimensionless shear rate λγ, (c) Dimensionless lanar elongational viscosity η EP /η 0 versus dimensionless lanar elongation rate λε. Fig. 5. Finite element meshes. 108

5 TANOUE NAGANAWA IEMOTO : Quasi-Three-Dimensional Simulation of Viscoelastic Flow through a Straight Channel with a Square Cross Section element meshes shown in Fig.5. The Weissenberg number We is defined as the following equation. The curve in the case using the 5 x 5 mesh is almost the same as that in the case using the 10 x 10 mesh. While, the curve in case using the 3 x 3 mesh is a little different from other two cases. As a result, we selected the 5 x 5 mesh in this study. The relationshi between the absolute value of the maximum dimensionless streamline function ψ max and the Weissenberg number We will be discussed in the next section. 3.2 The Effect of the Second Normal Stress Difference on the Secondary Flow (Case 1) First, we discuss the effect of the second normal stress difference N 2, i.e., Case 1 on the secondary flow. Figure 7 Fig. 6. (11) Comarison of the absolute value of maximum dimensionless streamline function ψ max as a function of We between the finite element meshes. The material constants of the test fluid are ε = 0.08, ξ = 0.2 and s = 1/9. shows the rojection of the streamlines of secondary flow at We = 1. The numerals in each figure are the dimensionless values of streamline function estimated by dividing the streamline function by <u>h. There are two secondary flows on the y-z lane in each case. Both secondary flows are counter-rotating each other. In articular, the fluid flows in a siral in the straight channel. The absolute value of the streamline function indicates the degree of secondary flow. It is shown that the secondary flow occurred in each case. In addition, the degree of secondary flow increases with ξ, i.e., the ratio between the second normal stress difference N 2 and the first one N 1, N 2 /N 1. It imlies that the degree of secondary flow deends on the absolute value of N 2 /N 1. For comarison, there is no secondary flow for the fluid without N 2, i.e., ξ = 0. Obviously, N 2 contributes to the secondary flow in the channel flow of the viscoelastic fluid. Let s discuss why the secondary flow direction was determined as Fig.7. Figure 8(a) shows the contour of dimensionless second normal stress difference (τ yy )/ (η 0 <u>/h). The absolute values of (τ yy <u>/h) in the region near the wall (the region of y/h > ca. 0.6 or z/h > ca. 0.6) are larger than those near the center and corner of the channel. This is because the shear rate in the region near the wall is larger than that near the center and corner of the channel. In addition, the (τ yy <u>/h) distribution is shown symmetrical with resect to the line from the center (y/h, z/h ) = (0, 0) to the corner (y/h, z/h) = (1, 1). Obviously, the two secondary flows would occur symmetrically to the line from (y/h, z/h) = (0, 0) to (y/h, z/h) = (1, 1). The negative value of (τ yy <u>/h) exists along the wall at y/h = 1. In this region near the wall at y/h = 1, the comressive stress in y direction is imosed on the fluid element. This fluid element would try to swell in y direction. This leads to the ressure near the center of the Fig. 7. Contours of the steamline for secondary flow at We = 1 in Case 1. Other materials constants are ε = 0.08 and s = 1/9. 109

6 Nihon Reoroji Gakkaishi Vol channel. We can also discuss the similar story about the influence of the normal stress difference (τ yy <u>/h) along the wall at z/h = 1 on the increase of ressure near the center of the channel. As the results, the ressure on the center of the channel becomes high (See Fig.8(b).). While, the ressure gradient along the y or z direction on the symmetric line (y/h = 0 or z/h = 0) is larger than that along the line from (y/h, z/h) = (0, 0) to (y/h, z/h) = (1, 1). Then the fluid flows from the center of the channel toward the wall at z/h = 1 or y/h = 1. For those reasons, the secondary flow occurs on the cross section of the channel as shown in Fig.7. Next we consider the effect of We on the secondary flow state in Case 1. Figure 9 shows the rojection of the streamline of secondary flow at various We. The material constants of focused fluid in these figures are ξ = 0.2, ε = 0.08 and s = 1/9, i.e., these are for the base fluid. The degree of secondary flow at We = 1 is larger than that at We = 0.2. However, the degree of secondary flow at We = 10 is smaller than that at We = 1 although Fig. 8. Contours of the dimensionless stress difference (τ yy -τ zz <u>/h) and ressure /(η 0 <u>/h) at We = 1 in Case 1. The material constants of the fluid are ξ = 0.2, ε = 0.08 and s = 1/9. the absolute value of the second normal stress difference N 2 increases with shear rate, i.e., We. In order to discuss the degree of secondary flow quantitatively, Fig.10 shows the relationshi between ψ max and We. Because the large value of ψ max indicates the high ercentage of secondary flow velocity to main flow velocity, ψ max corresonds to the degree of secondary flow. The value of ψ max increases with ξ in the low We region (We < about 10). This tendency is the same as the calculation results by Xue et al. 4) In the high We region (We > about 10), it can be judged that ψ max is almost indeendent of ξ although ψ max at ξ = 0.05 is the largest at the same We. However, the value of ψ max decreases with increasing We at each material constant ξ in this region (We > about 10) although the contribution of N 2 increases with ξ. Therefore, other roerties may contribute to the secondary flow. We focused the recoverable shear for the second normal stress difference defined as N 2 /2σ, where σ is a shear stress. Fig. 9. Contours of the steamline of secondary flow at various We. The material constants of the fluid are ξ = 0.2, ε = 0.08 and s = 1/9. 110

7 TANOUE NAGANAWA IEMOTO : Quasi-Three-Dimensional Simulation of Viscoelastic Flow through a Straight Channel with a Square Cross Section The reason is that the shear viscosity would make some influences on the secondary flow in addition to N 2. Figure 11 shows the relationshi between N 2 /2σ and dimensionless shear rate λγ for the PTT model in Case 1. This can be calculated by using the steady-state shear flow characteristics shown in Fig.3. In the low shear rate region ( λγ < about 3 10), N 2 /2σ increases with λγ. However, N 2 /2σ shows the maximum at λγ = about 3 10 and decreases with λγ in the high shear rate region ( λγ > about 3 10). The shaes of the curves in Fig.11 are almost similar to those in Fig.10 for the same fluid although We at ψ max is different from λγ at the maximum N 2 /2σ. In addition, We at ψ max is roortional to λγ at the maximum N 2 /2σ. These imly that the degree of secondary flow would be related to not only N 2 but also N 2 /2σ. We calculated the average value of dimensionless shear rare on y-z lane at We that is the case of ψ max. The shear rate γ at each node on y-z lane is calculated using the second R invariant of the rate of deformation tensor. Here, We and at ψ max and λγ at the maximum N 2 /2σ are called We max, max and λγ max, resectively. In case of ξ = 0.05, We max = 5, max = 12 and λγ max = 13. In case of ξ = 0.2, We max = 1.5, max = 3.6 and λγ max = 5.6. In case of ξ = 0.5, We max = 0.8, = 2.0 and max λγ max = 3.6. In case of same ξ, is not max equal to λγ max. However, roortionally increases with max λγ max. This imlies that N 2 /2σ is one of the main factors for secondary flow although there are other factors for it. 3.3 The Effect of the Elongational Flow Characteristics on the Secondary Flow (Case 2) Next, we discuss the effect of the elongational flow characteristics, i.e., Case 2 on the secondary flow. Figure 12 shows the relationshi between ψ max and We. In the low We region (We < ca. 0.4), ψ max for the fluid with ε = is the smallest of all three test fluids. In the region of We > 0.4, ψ max for the fluid with ε = is almost the same as that for the fluid with ε = While, ψ max for the fluid with ε = 0.4 is the smallest of all three test fluids in the region of 0.4 < We < 2, and ψ max for the fluid with ε = 0.4 is the largest of all three test fluids in the region of We > 2. Each curve has the maximum oint, i.e., the degree of secondary flow decreases with an increase of We in the high We region. This tendency is similar to that in Case 1. Figure 13 shows the relationshi between N 2 /2σ and λγ for the PTT model in Case 2. This can be calculated by using the steady-state shear flow characteristics Fig. 10. The absolute value of maximum dimensionless streamline function ψ max as a function of We in Case 1. Other material constants are ε = 0.08 and s = 1/9. Fig. 12. The absolute value of maximum dimensionless streamline function ψ max as a function of We in Case 2. Other material constants are ξ = 0.2 and s = 1/9. Fig. 11. The recoverable shear for the second normal stress difference N 2 /2σ as a function of dimensionless shear rate λγ for the PTT model in Case 1. Fig. 13. The recoverable shear for the second normal stress difference N 2 /2σ as a function of dimensionless shear rate λγ for the PTT model in Case

8 Nihon Reoroji Gakkaishi Vol shown in Fig.4. The shaes of these curves are almost similar to that in Case 1 shown in Fig.11. This imlies that N 2 /2σ contributes to the degree of secondary flow. However the relationshi between N 2 /2σ and ψ max may not be totally corresonding. For examle, ψ max of the test fluid for the fluid with ε = 0.4 is the largest of all three fluids in We > ca. 2 although N 2 /2σ for the test fluid with ε = 0.4 is almost the same as other test fluids in the high shear rate region. This indicates that the elongational flow characteristics would make some influences of the secondary flow state. In order to consider quantitatively the effect of elongational flow characteristics on the secondary flow state, we consider the contour of the absolute value of lanar elongation rate on y-z lane. The absolute value of lanar elongation rate ε is defined as the follow equation. (12) We can consider the total value of lanar elongation rate on the secondary flow by use of ε. Figure 14 shows the contour of dimensionless lanar elongation rate λε on y-z lane for the test fluids at We = 1 and 100 in Case 2. Some elongational flow occurs along the stremaline on the secondary flow. The values of λε on y-z lane at We = 1 are lower than about 0.05 for each test fluid. The maximum value of λε on y-z lane for the test fluid with ε = and 0.08 is larger than that for the test fluid with ε = 0.4. On the other hand, the maximum value of λε on y-z lane for the test fluid with ε = and 0.08 is smaller than that for the test fluid with ε = 0.4 although the values of λε at We = 100 are lower than about 0.16 for each test fluid. According to the lanar elongational flow characteristics shown in Fig.15, the test fluids with ε = and 0.08 show the strong strain-hardening character in λε > ca It would not be easy to occur the accerelation flow in the region of λε > ca for the test fluid with ε = and This is one reason why the degree of secondary flow for the fluid with ε = 0.4 is the largest of all three test fluids at the same We in the high We region. However, the contribution of elongational flow characteristics to the secondary flow is smaller than that of second normal stress difference, i.e., the recoverable shear for second normal stress difference N 2 /2σ. Fig. 14. Contours of dimensionless lanar elongation rate λε on y-z lane in Case 2. The other material constants are ξ = 0.2 and s = 1/9. 112

9 TANOUE NAGANAWA IEMOTO : Quasi-Three-Dimensional Simulation of Viscoelastic Flow through a Straight Channel with a Square Cross Section to the secondary flow state. On the whole, the degree of secondary flow would increase with the strain-hardening. However, the contribution of elongational flow characteristics to the secondary flow is smaller than that of the recoverable shear N 2 /2σ. Fig. 15. Dimensionless lanar elongational viscosity η EP /η 0 versus dimensionless lanar elongation rate λε in low region in Case 2. λε 4. CONCLUSIONS We discussed the effect of flow characteristics on secondary flow states of the viscoelastic fluid through a straight channel with a square cross section by the viscoelastic flow simulation using the finite element method. The Phan-Thien Tanner (PTT) model was emloyed as the constitutive equation. The second normal stress difference N 2 contributes to the secondary flow of the viscoelastic fluid through a straight channel with a square cross section. The degree of secondary flow exressed by the absolute value of maximum streamline function ψ max deends on the recoverable shear for the second normal stress difference N 2 /2σ. In addition, according to the consideration of the distribution of N 2 on the cross section, the flow direction of secondary flow deends on the ressure dro generated by N 2. The elongational flow characteristics would also contribute REFERENCES 1) Townsend P, Walters K, Waterhouse WM, J Non-Newton Fluid Mech, 1, 176 (1976). 2) Gervang B, Larsen PS, J Non-Newton Fluid Mech, 39, 217 (1991). 3) Criminale WO, Ericksen JL, Filbey GL, Arch Rational Mech Anal, 1, 410 (1958). 4) Xue SC, Phan-Thien N, Tanner RI, J Non-Newton Fluid Mech, 59, 191 (1995). 5) Phan-Thien N, Tanner RI, J Non-Newton Fluid Mech, 2, 353 (1977). 6) Debbaut B, Avalosse T, Dooley J, Hughes K, J Non-Newton Fluid Mech, 69, 255 (1997). 7) Debbaut B, Dooley J, J Rheol, 43, 1525 (1998). 8) Giesekus H, J Non-Newton Fluid Mech, 11, 69 (1982). 9) Torres A, Hrymak AN, Vlachooulus J, Rheol Acta, 32, 513 (1993). 10) Gifford WA, Polym Engng Sci, 37, 315 (1997). 11) Dooley J, Hyun KS, Hughes K, Polym Engng Sci, 38, 1060 (1998). 12) Takase M, Kihara S, Funatsu K, Rheol Acta, 37, 634 (1998). 13) Sunwoo KB, Park SJ, Lee SJ, Ahn KH, Lee SJ, Rheol Acta, 41, 144 (2002). 14) Tanoue S, Kajiwara T, Iemoto Y, Funatsu K, Polym Engng Sci, 38, 409 (1998). 113